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List Of Long Mathematical Proofs
This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable. , the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full. Long proofs The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. *1799 The AbelâRuffini theorem was nearly proved by Paolo Ruffini, but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel published a proof that required just six pages. *1890 Killing's classification of simple complex Lie algebras, including his discovery of the exceptional Lie algebras, took 180 pages in 4 papers. *1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ĂlĂ©ments De GĂ©omĂ©trie AlgĂ©brique
The (''EGA''; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the . In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation and basic reference of modern algebraic geometry. Editions Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the '' Séminaire de géométrie algébrique'' (known as ''SGA''). Indeed, as explained by Grothendieck in the preface of the published version of ''SGA'', by 1970 it had become clear that incorporating all of the planned material in ''EGA'' would require significan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fondements De La Géometrie Algébrique
''Fondements de la GĂ©ometrie AlgĂ©brique'' (''FGA'') is a book that collected together seminar notes of Alexander Grothendieck. It is an important source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments. The title is a translation of the title of AndrĂ© Weil's book ''Foundations of Algebraic Geometry.'' It contained material on descent theory, and existence theorems including that for the Hilbert scheme. The ''Technique de descente et thĂ©orĂšmes d'existence en gĂ©ometrie algĂ©brique'' is one series of seminars within ''FGA''. Like the bulk of Grothendieck's work of the IHĂS period, duplicated notes were circulated, but the publication was not as a conventional book. Contents These are SĂ©minaire Bourbaki notes, by number, from the years 1957 to 1962.Fondements de la gĂ©omĂ©trie algĂ©brique. Commentaires Ă©minaire Bourbaki, t. 14, 1961/62, ComplĂ©mentThĂ©orĂšme de dualitĂ© pour les faisceaux algĂ©br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponent Of A Group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent that divides its order. Infinite examples Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the PrĂŒfer groups. Another example is the direct sum of all dihedral groups. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich (see '' GolodâShafarevich theorem''), and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z. ** A locally cyclic group is a group in which every finitely gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside's Problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory, and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below). Some of these variants are still open questions. Brief history Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Adian
Sergei Ivanovich Adian, also Adyan (; ; 1 January 1931 â 5 May 2020), 4381, and hence for all multiples of those odd integers as well. The solution of the Burnside problem was certainly one of the most outstanding and deep mathematical results of the past century. At the same time, this result is one of the hardest theorems: just the inductive step of a complicated induction used in the proof took up a whole issue of volume 32 of Izvestiya, even lengthened by 30 pages. In many respects the work was literally carried to its conclusion by the exceptional persistence of Adian. In that regard it is worth recalling the words of Novikov, who said that he had never met a mathematician more âpenetratingâ than Adian. In contrast to the AdianâRabin theorem, the paper of Adian and Novikov in no way âclosedâ the Burnside problem. Moreover, over a long period of more than ten years Adian continued to improve and simplify the method they had created and also to adapt the method for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyotr Novikov
Pyotr Sergeyevich Novikov (; 15 August 1901, Moscow â 9 January 1975, Moscow) was a Soviet mathematician known for his work in group theory. His son, Sergei Novikov, was also a mathematician. Early life and education Pyotr Sergeyevich Novikov was born on 15 August 1901 in Moscow, Russia to Sergei Novikov, a merchant, and Alexandra Novikov. He served in the Red Army during the Russian Civil War from 1920 to July 1922. He studied at Moscow University from 1919 to 1920 and again from 1922 until he graduated in 1925. He studied under Nikolai Luzin until he finished his graduate studies in 1929. Career Novikov worked at the Moscow D. Mendeleev Institute of Chemical Technology from 1929 until 1934, when he joined the Department of Real Function Theory at the Steklov Institute of Mathematics. He was awarded his doctorate in 1935 and promoted to full professor in 1939. Novikov became head of the Department of Analysis at the Moscow State Teachers Training Institute in 1944. In 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Series Representation
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on LÂČ(''G''). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation. Properties If ''G'' is unimodular group, unimodular, an irreducible unitary representation Ï of ''G'' is in the discrete series if and only if one (and hence all) matrix coefficient :\langle \rho(g)\cdot v, w \rangle \, with ''v'', ''w'' non-zero vectors is square-integrable on ''G'', with respect to Haar measure. When ''G'' is unimodular, the discrete series representation has a formal dimension ''d'', with the property that :d\int \langle \rho(g)\cdot v, w \rangle \overlinedg =\langle v, x \rangle\overline for ''v'', ''w'', ''x'', ''y'' in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
